# Bayesian System ID: Optimal management of parameter, model, and measurement uncertainty

arXiv: https://arxiv.org/abs/2003.02359 · HTML: https://mlsanity.com/paper/2003.02359/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Mar 2020.

## Built on by (most cited first, 16 of 16)

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- [Bayesian Identification of Nonseparable Hamiltonian Systems Using Stochastic Dynamic Models](https://mlsanity.com/paper/2209.07646/) (2022)
- [Sparse Identification of Nonlinear Dynamics with Conformal Prediction](https://mlsanity.com/paper/2507.11739/) (2025)
- [Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modeling](https://mlsanity.com/paper/2401.12476/) (2024)
- [Active search for Bifurcations](https://mlsanity.com/paper/2406.11141/) (2024)
- [Discovering Phase Space Structure in Learned Hamiltonian Systems](https://mlsanity.com/paper/2509.23003/) (2025)
- [Data-driven sparse identification of governing PDEs via knockoff filters and multi-criteria trade-offs](https://mlsanity.com/paper/2605.26631/) (2026)
- [Likelihood-based generalization of Markov parameter estimation and multiple shooting objectives in system identification](https://mlsanity.com/paper/2212.13902/) (2022)
- [Canonical Bayesian Linear System Identification](https://mlsanity.com/paper/2507.11535/) (2025)
- [A Weak Penalty Neural ODE for Learning Chaotic Dynamics from Noisy Time Series](https://mlsanity.com/paper/2511.06609/) (2025)
- [Frequency-Separable Hamiltonian Neural Network for Multi-Timescale Dynamics](https://mlsanity.com/paper/2603.06354/) (2026)
- [Nonlinear Bayesian Estimator for Parameter Learning: A Fixed-Point Characterization](https://mlsanity.com/paper/2606.10111/) (2026)

## Abstract (arXiv)

We evaluate the robustness of a probabilistic formulation of system identification (ID) to sparse, noisy, and indirect data. Specifically, we compare estimators of future system behavior derived from the Bayesian posterior of a learning problem to several commonly used least squares-based optimization objectives used in system ID. Our comparisons indicate that the log posterior has improved geometric properties compared with the objective function surfaces of traditional methods that include differentially constrained least squares and least squares reconstructions of discrete time steppers like dynamic mode decomposition (DMD). These properties allow it to be both more sensitive to new data and less affected by multiple minima --- overall yielding a more robust approach. Our theoretical results indicate that least squares and regularized least squares methods like dynamic mode decomposition and sparse identification of nonlinear dynamics (SINDy) can be derived from the probabilistic formulation by assuming noiseless measurements. We also analyze the computational complexity of a Gaussian filter-based approximate marginal Markov Chain Monte Carlo scheme that we use to obtain the Bayesian posterior for both linear and nonlinear problems. We then empirically demonstrate that obtaining the marginal posterior of the parameter dynamics and making predictions by extracting optimal estimators (e.g., mean, median, mode) yields orders of magnitude improvement over the aforementioned approaches. We attribute this performance to the fact that the Bayesian approach captures parameter, model, and measurement uncertainties, whereas the other methods typically neglect at least one type of uncertainty.